422 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XVIII A. Desboves125 verified that every even number between 2 and 10000 is a sum of two primes in at least two ways; while, if the even number is the double of an odd number, it is simultaneously a sum of two primes of the form 4n+l and also a sum of two primes of the form 4n— 1. J. J. Sylvester126 stated that the number of ways of expressing a very-large even number n as a sum of two primes is approximately the ratio of the square of the number of primes <n to n, and hence bears a finite ratio to the quotient of n by the square of the natural logarithm of n. [Cf. Stackel132]. F. J. E. Lionnet127 designated by x the number of ways 2a can be expressed as a sum of two odd primes, by y the number of ways 2a can be expressed as a sum of two distinct odd composite numbers, by z the number of odd primes <2a, and by q the largest integer ^o/2. He proved that q+x—y+z and argued that it is very probable that there are values of n for which q^y+z, whence x = 0. N. V. Bougaief m° noted that, if M (n) denotes the number of ways n can be expressed as a sum of two primes, and if ^ denotes the ith prime >1, G. Cantor128 verified Goldbach's theorem up to 1000. His table gives the number of decompositions of each even number < 1000 as a sum of two primes and lists the smaller prime. V. Aubry129 verified the theorem from 1002 to 2000. R. Haussner130 verified the law up to 10000 and announced results observed by a study of his131 tables up to 5000. His table I (pp. 25-178) gives the number v of decompositions of every even n up to 3000 as a sum x+y of two primes and the values of x (x^y), as in the table by Cantor. His table II (pp. 181-191) gives v for 2<n<5000; this table and further computations enable him to state that Goldbach's theorem is true for n< 10000. Let P(2p+l) be the number of all odd primes 1, 3, 5, ... which are ^2p+l, and set £(2p+l)=P(2p+l)-2P(2p-l)+P(2p-3), P(-1)=P(-3)=0. Then the number of decompositions of 2n into a sum of two primes z, y p.O If € = 1 or —1 according as n is a prune or not, P-I 2 mNouv. Ann. Math., 14, 1855, 293. 126Proc. London Math. Soc., 4, 1871-3, 4-6; Coll. M. Papers, 2, 709-711. W7Nouv, Ann. Math., (2), 18, 1879, 356. Cf . Assoc. frang. av. sc., 1894, I, p. 96. »M3ompte8 Rendus Paris, 100, 1885, 1124. "•Assoc. franc.. av. sc., 1894, 117-134; PintermeMiaire des math., 2, 1895, 179. U9L'interme"diaire des math., 3, 1896, 75; 4, 1897, 60; 10, 1903, 61 (errata, p. 166, p. 283). wojahresbericht Deutschen Math.-Verein., 5, 1896, 62-66. Verhandlungen Gesell. Deutscher Naturforscher u. Aerzte, 1896, II, 8. 181Nova Acta Acad. Caes. Leop.-Carolinae, 72, 1899, 1-214.